We have the following qualitative result about the topological entropy of the maps .
For p=1 or , the curve is a square and is integrable with integral I(s,y)=y+y', where . The topological entropy is then zero. For p=2, the curve is a circle and is integrable having the integral I(s,y)=y. On each invariant curve I=const, the map is a rotation. The topological entropy is also zero.
For p;SPMgt;2, there exist points on the curve where the curvature is zero. A theorem of Mather  assures that there exists then no homotopiclly nontrivial invariant curve. Angenent's result  implies then that the topological entropy is non-vanishing. For p;SPMlt;2, there exist points on the table where the curvature is unbounded. A theorem of Hubacher  assures that there exist no invariant curves near the boundary of X. Again, Angenent's result leads to positive topological entropy.
End of the proof.